PostgreSQL is an open-source object-relational database system that uses and extends SQL. The project began in 1986 as POSTGRES at the University of California, Berkeley, and runs on all major operating systems. Among other things it offers ACID transactions, JSON/JSONB, a range of index types and parallel queries, replication and point-in-time recovery, row- and column-level security, and extensions such as PostGIS for geospatial data. You download and operate PostgreSQL yourself on your own servers or in your own containers.
PostgreSQL is developed by a worldwide community of contributors under the name PostgreSQL Global Development Group. It is released under the PostgreSQL Licence, a liberal open-source licence similar to BSD and MIT, and the developers state they have no plans to change it. Because you host it yourself, you decide where the data resides and which jurisdiction applies. For a European organisation, self-hosting on its own or European infrastructure is therefore a straightforward route to control over its data.